I've started teaching a few classes for underprivileged kids. I have a shoe-string budget to work with, most of the money is going to rent instruments. Where can I find affordable music books for beginners?

My go-to answer whenever someone asks where they can get something "affordable" is Amazon. I would definitely check out there! If you have any local used book shops around you, they might also have something there for you. You could also opt for the non-book method and just print out free material you find on the internet, although I'm not sure if that would be more price-efficient in the long run or not. How many students do you have?

I've previously found affordable beginner music books on ebay.
If you don't mind buying used books, you might try there.
A lot of classical music has entered the public domain. The music composition itself is free to use. Take a look at this non-profit which has free textbooks and sheet music. https://musopen.org/

Lost the Remote is right. If you're looking for classical music you should be able to find free resources all around the web. Other than the great suggestions already mentioned, you could also check out sites like Half Price Books.

Have you ever heard of 8notes? They have a lot of free sheet music. If you get a subscription, you get unlimited downloads, unlimited playalong tracks, and some other fun stuff. You can read more about the benefits here. It's only $20 a year, so I don't imagine it gets much cheaper than that.

You could also try advertising in your local newspaper or Craiglist. There are tons of people who hold onto books they never use anymore because they don't like the idea of giving them away and ending up in the trash. I know it's weird, but it's true. I can speak from first-hand experience. People might be willing to give you some secondhand books for cheap, maybe even free!